Question

suppose the labor cost (in dollars) for manufacturing a camera can be approximated by
l(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y - 2xy + 112
where ( x ) is the number of hours required by a skilled craftsperson and ( y ) is the number of hours required by a semiskilled person. find values of ( x ) and ( y ) that minimize the labor cost. find the minimum labor
cost.
labor cost will be minimized when ( x=square ) and ( y=square ).

Explanation:

Step1: Find the partial derivatives

Find \(L_x\) and \(L_y\).
For \(L(x,y)=\frac{3}{2}x^{2}+y^{2}-6x - 4y-2xy + 112\),
\(L_x = 3x-6 - 2y\) (using power rule \((x^n)^\prime=nx^{n - 1}\))
\(L_y=2y-4 - 2x\)

Step2: Set the partial derivatives equal to zero

Set \(L_x = 0\) and \(L_y = 0\).
\(

$$\begin{cases}3x-2y-6 = 0\\-2x + 2y-4=0\end{cases}$$

\)
Add the two equations: \((3x-2y-6)+(-2x + 2y-4)=0\), which simplifies to \(x-10 = 0\), so \(x = 10\)
Substitute \(x = 10\) into \(3x-2y-6 = 0\), we get \(3\times10-2y-6 = 0\), \(30-2y-6=0\), \(24-2y = 0\), \(y = 12\)

Step3: Check the second - derivative conditions

Find \(L_{xx}=3\), \(L_{xy}=-2\), \(L_{yy}=2\)
The discriminant \(D = L_{xx}L_{yy}-(L_{xy})^{2}=3\times2-(-2)^{2}=6 - 4=2>0\) and \(L_{xx}=3>0\)

Answer:

\(x = 10\) and \(y = 12\)